3.371 \(\int x^3 \left (a+b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]

[Out]

-(a*(a + b*x^2)^(5/2))/(5*b^2) + (a + b*x^2)^(7/2)/(7*b^2)

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Rubi [A]  time = 0.0657783, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+b x^2\right )^{7/2}}{7 b^2}-\frac{a \left (a+b x^2\right )^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^(3/2),x]

[Out]

-(a*(a + b*x^2)^(5/2))/(5*b^2) + (a + b*x^2)^(7/2)/(7*b^2)

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Rubi in Sympy [A]  time = 7.91357, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**(3/2),x)

[Out]

-a*(a + b*x**2)**(5/2)/(5*b**2) + (a + b*x**2)**(7/2)/(7*b**2)

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Mathematica [A]  time = 0.0316473, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{5/2} \left (5 b x^2-2 a\right )}{35 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^(3/2),x]

[Out]

((a + b*x^2)^(5/2)*(-2*a + 5*b*x^2))/(35*b^2)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-5\,b{x}^{2}+2\,a}{35\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^(3/2),x)

[Out]

-1/35*(b*x^2+a)^(5/2)*(-5*b*x^2+2*a)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234097, size = 61, normalized size = 1.61 \[ \frac{{\left (5 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + a^{2} b x^{2} - 2 \, a^{3}\right )} \sqrt{b x^{2} + a}}{35 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="fricas")

[Out]

1/35*(5*b^3*x^6 + 8*a*b^2*x^4 + a^2*b*x^2 - 2*a^3)*sqrt(b*x^2 + a)/b^2

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Sympy [A]  time = 3.44814, size = 85, normalized size = 2.24 \[ \begin{cases} - \frac{2 a^{3} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{a^{2} x^{2} \sqrt{a + b x^{2}}}{35 b} + \frac{8 a x^{4} \sqrt{a + b x^{2}}}{35} + \frac{b x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**(3/2),x)

[Out]

Piecewise((-2*a**3*sqrt(a + b*x**2)/(35*b**2) + a**2*x**2*sqrt(a + b*x**2)/(35*b
) + 8*a*x**4*sqrt(a + b*x**2)/35 + b*x**6*sqrt(a + b*x**2)/7, Ne(b, 0)), (a**(3/
2)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.212545, size = 105, normalized size = 2.76 \[ \frac{\frac{7 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}}{b}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^3,x, algorithm="giac")

[Out]

1/105*(7*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*a/b + (15*(b*x^2 + a)^(7/
2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)/b)/b